Library / Dimensional analysis and similarity / DIM-20
Dimensional analysis and similarity
Pump or fan shaft power at a new speed, diameter and density
Affinity law for shaft power: for geometrically similar machines at corresponding operating points and comparable efficiency, power is proportional to density, the cube of the speed and the fifth power of the diameter.
- Ref
- DIM-20
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- P2 [kW]
With s = n2/n1, d = D2/D1 and z = ρ2/ρ1, the shaft power at the corresponding operating point scales as P2 = P1 z s³ d⁵. It assumes comparable efficiency at the two points. These affinity laws compare corresponding operating points of geometrically similar rotodynamic machines. They assume low-compressibility fan flow or non-cavitating pump flow. The diameter powers describe a fully scaled machine; trimming an impeller in an existing casing changes the geometry ratios and needs a trim curve or manufacturer data.
Affinity law for shaft power: for geometrically similar machines at corresponding operating points and comparable efficiency, power is proportional to density, the cube of the speed and the fifth power of the diameter.
Calculator
- Speed ratio, s1.2
- Diameter ratio, d1
- Power multiplier, z s³ d⁵1.728
- Power if the new speed is reduced by 20 %5.308 kW
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Raising the speed by a factor of 1.2 (with the same diameter and fluid) multiplies the power by 1.728, from 6 to 10.37 kW: a 20 % increase in speed costs 73 % more power.
- The cube law cuts both ways: slowing a pump or fan by 20 % cuts its power to about half, which is why variable-speed drives save so much energy in part-load duty.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kW) | Local step | Change in result |
|---|---|---|---|---|---|---|
| D1 Impeller diameter at point 1 | 20 to 3,000 mm | 1.037e-4 | 7,873,000 | 7,873,000 | +10% | −37.9 % |
| D2 Impeller diameter at point 2 | 20 to 3,000 mm | 1.365e-5 | 1,037,000 | 1,037,000 | +10% | +61.1 % |
| ρ1 Fluid density at point 1 | 0.1 to 2,000 kg/m³ | 5.184 | 103,700 | 103,700 | +10% | −9.09 % |
| n1 Speed at point 1 | 50 to 10,000 rpm | 0.01037 | 82,940 | 82,940 | +10% | −24.9 % |
| P1 Shaft power at point 1 | 0.01 to 5,000 kW | 0.01728 | 8,640 | 8,640 | +10% | +10 % |
| n2 Speed at point 2 | 50 to 10,000 rpm | 7.5e-4 | 6,000 | 6,000 | +10% | +33.1 % |
| ρ2 Fluid density at point 2 | 0.1 to 2,000 kg/m³ | 0.001037 | 20.74 | 20.73 | +10% | +10 % |
Sweep of Speed at point 2
| Speed at point 2 (rpm) | Shaft power at point 2 (kW) |
|---|---|
| 50 | 7.5e-4 |
| 1,294 | 12.99 |
| 2,538 | 98.03 |
| 3,781 | 324.4 |
| 5,025 | 761.3 |
| 6,269 | 1,478 |
| 7,513 | 2,544 |
| 8,756 | 4,028 |
| 10,000 | 6,000 |
Sweep one variable
Variables and how each one contributes
Shaft power at point 2
What the equation gives you. Everything else feeds it.
Shaft power at point 1
The shaft power at the original operating point, in kW. Raises the new power in direct proportion. Raising it by 10 % (to 6.6) moves the result by +10 %, from 10.37 to 11.4 kW.
Speed at point 1
The original rotational speed, in rpm. Lowers the new power with the cube of the speed ratio. Raising it by 10 % (to 1,100) moves the result by −24.9 %, from 10.37 to 7.79 kW.
Speed at point 2
The new rotational speed, in rpm. Raises the new power with the cube of the speed, so speed is a strong lever. Raising it by 10 % (to 1,320) moves the result by +33.1 %, from 10.37 to 13.8 kW.
Impeller diameter at point 1
The original impeller diameter, in mm. Lowers the new power with the fifth power of the diameter ratio. Raising it by 10 % (to 330) moves the result by −37.9 %, from 10.37 to 6.438 kW.
Impeller diameter at point 2
The new impeller diameter, in mm. Raises the new power with the fifth power of the diameter, so diameter is the strongest lever. Raising it by 10 % (to 330) moves the result by +61.1 %, from 10.37 to 16.7 kW.
Fluid density at point 1
The fluid density at the original point, in kg/m³. Lowers the new power in inverse proportion. Raising it by 10 % (to 1,100) moves the result by −9.09 %, from 10.37 to 9.425 kW.
Fluid density at point 2
The fluid density at the new point, in kg/m³. Raises the new power in direct proportion. Raising it by 10 % (to 1,100) moves the result by +10 %, from 10.37 to 11.4 kW.
Worked example with the values above
- Ratios: s = 1.2, d = 1 and z = 1.
- Power multiplier: z s³ d⁵ = 1 × 1.2³ × 1⁵ = 1.728.
- P2 = 6 × 1.728 = 10.368 kW.
Assumptions and limits
- Geometric similarity and corresponding operating points, with comparable efficiency at the two points.
- Low-compressibility fan flow or non-cavitating pump flow.
Common mistakes
- Sizing the motor for the scaled power without allowing for a change in efficiency or a margin.
- Applying the fifth-power diameter law to a trimmed impeller.
- Using the electrical input power at one point and the shaft power at the other.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (P1 × 1, ρ2 × 1, n2 × 3, n1 × −3, D2 × 5, D1 × −5). The diameter (fifth power) and speed (cube) errors dominate, and efficiency changes between the two points add a few per cent on top of that.
Where it is used
- Estimating the power a pump or fan needs after a speed, size or density change.
- Estimating the energy saved by running a pump or fan more slowly.
Maths you need
- Shaft power: power delivered to the impeller by the shaft
- Efficiency: useful power divided by shaft power